Left and Right Uniformities Restricted to Subspaces of a Topological Group
by
Michael G. Tkačenko
Metropolitan Autonomous University

For a topological group G, denote by *V_G and V*_G the left and right group uniformities of G respectively. If X is a subspace of G then the uniformities U_l = *V_G |_X and U_r = V*_G |_X are concordant in the sense that they both induce on X the topology X inherits from G. We consider the following problem.

Problem. Let U_l and U_r be concordant uniformities on a Tikhonov space X. Does there exist a Hausdorff topological group G containing X as a subspace such that *V_G |_X = U_l and V*_G |_X = U_r?

We show that in general the answer to this problem is "no" by constructing two concordant uniformities on a countable discrete space X which can not be simultaneosly generated by any embedding of X into a topological group.

Let a uniformity U on a zero-dimensional space X be generated by a family \Gamma of disjoint open covers. Denote by U_fin the uniformity for X generated by all finite covers belonging to \Gamma.

Theorem. Let U_l and U_r be concordant zero-dimensional uniformities on a completely regular space X and suppose that U_lfin = (U_r)_fin. Then there exists a Hausdorff topological group G containing X as a closed subspace such that *V_G |_X = U_l and V*_G |_X = U_r.

Date received: April 12, 1996

Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caaf-83.